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Abelian Varieties

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.003.0009
Subject:
Mathematics, Geometry / Topology

Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this chapter ... More


Fourier-Mukai Transforms in Algebraic Geometry

D. Huybrechts

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.001.0001
Subject:
Mathematics, Geometry / Topology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this ... More


Hodge Structures with Complex Multiplication

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691154244
eISBN:
9781400842735
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691154244.003.0006
Subject:
Mathematics, Analysis

This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining ... More


Structures of Semi-Abelian Schemes

Kai-Wen Lan

in Arithmetic Compactifications of PEL-Type Shimura Varieties

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691156545
eISBN:
9781400846016
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691156545.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups ... More


Theory of Degeneration for Polarized Abelian Schemes

Kai-Wen Lan

in Arithmetic Compactifications of PEL-Type Shimura Varieties

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691156545
eISBN:
9781400846016
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691156545.003.0004
Subject:
Mathematics, Geometry / Topology

This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over ... More


Mordell-Weil Groups and Generating Series

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

in The Gross-Zagier Formula on Shimura Curves

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691155913
eISBN:
9781400845644
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691155913.003.0003
Subject:
Mathematics, Number Theory

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before ... More


Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691156545
eISBN:
9781400846016
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691156545.001.0001
Subject:
Mathematics, Geometry / Topology

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical ... More


On the Chow ring of <i>K3</i> surfaces and hyper-Kahler manifolds

Claire Voisin

in Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160504
eISBN:
9781400850532
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160504.003.0005
Subject:
Mathematics, Geometry / Topology

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was ... More


Introduction and Statement of Main Results

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

in The Gross-Zagier Formula on Shimura Curves

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691155913
eISBN:
9781400845644
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691155913.003.0001
Subject:
Mathematics, Number Theory

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves ... More


Trace of the Generating Series

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

in The Gross-Zagier Formula on Shimura Curves

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691155913
eISBN:
9781400845644
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691155913.003.0004
Subject:
Mathematics, Number Theory

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the ... More


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